Development of the Airline Business Macroeconomics Dynamics Models

In the presented problem setting, described with Eqs (1) and (2), determination of the objective functional for the individuals’ subjective preferences π is given by 3 Φπ=−∑i=1N=3πtilnπti+βππtKKt+πtHHt+πtCCt+γ∑i=1N=3π1i−1, \[{{\text{ }\!\!\Phi\!\!\text{ }}_{i}}=-\underset{i=1}{\overset{N=3}{\mathop \sum }}\,\pi _{t}^{i}\ln \pi _{t}^{i}+{{\beta }_{\pi }}\left[ \pi _{t}^{K}{{K}_{t}}+\pi _{t}^{H}{{H}_{t}}+\pi _{t}^{C}{{C}_{t}} \right]+\gamma \left[ \underset{i=1}{\overset{N=3}{\mathop \sum }}\,\pi _{1}^{i}-1 \right],\] where the uncertainty of the available alternatives is evaluated with the subjective entropy member 4 −∑i=1N=3πtilnπti; \[-\underset{i=1}{\overset{N=3}{\mathop \sum }}\,\pi _{1}^{i}\,\ln \,\pi _{t}^{i};\] β_π and γ are the corresponding cognitive coefficients.

The normalising condition is expressed with the member of 5 ∑i=1N=3πti−1. \[\left[ \underset{i=1}{\overset{N=3}{\mathop \sum }}\,\pi _{1}^{i}-1 \right].\]

The necessary conditions for the objective functional in Eq. (3)’s extremum existence 6 ∂Φπ∂πti=0, \[\frac{\partial {{\text{ }\!\!\Phi\!\!\text{ }}_{\pi }}}{\partial \pi _{t}^{i}}=0,\] allow obtaining an optimal distribution of the individuals’ subjective preferences π, as follows: 7 πtK,πtHandπtC. \[\pi _{t}^{K},\pi _{t}^{H}\,\text{and}\,\,\pi _{t}^{C}.\]

That is, 8 −lnπtK−1+βπKt+γ=0. \[-\ln \,\pi _{t}^{K}-1+{{\beta }_{\pi }}{{K}_{t}}+\gamma =0.\]

This yields 9 lnπtK=γ−1+βπKt. \[\ln \,\pi _{t}^{K}=\gamma -1+{{\beta }_{\pi }}{{K}_{t}}.\]

Thus, 10 πtK=eγ−1+βπKt. \[\pi _{t}^{K}={{e}^{\gamma -1+{{\beta }_{\pi }}{{K}_{t}}}}.\]

On the other hand 11 −lnπtH−1+βπHt+γ=0. \[-\ln \pi _{t}^{H}-1+{{\beta }_{\pi }}{{H}_{t}}+\gamma = 0.\]

This yields 12 lnπtH=γ−1+βπHt. \[\ln \pi _{t}^{H}=\gamma -1+{{\beta }_{\pi }}{{H}_{t}}.\]

Thus, 13 πtH=eγ−1+βπHt. \[\pi _{t}^{H}={{e}^{\gamma -1+{{\beta }_{\pi }}{{H}_{t}}}}.\]

And, 14 −lnπtC−1+βπCt+γ=0. \[-\ln \pi _{t}^{C}-1+{{\beta }_{\pi }}{{C}_{t}}+\gamma =0.\]

This yields 15 lnπtC=γ−1+βπCt. \[\ln \pi _{t}^{C}=\gamma -1+{{\beta }_{\pi }}{{C}_{t}}.\]

Thus, 16 πtC=eγ−1+βπCt. \[\pi _{t}^{C}={{e}^{\gamma -1+{{\beta }_{\pi }}{{C}_{t}}}}.\]

The procedure of Eqs (3)–(16) leads to the following position: 17 πtK+πtH+πtC=eγ−1+βπKt+eγ−1+βπHt+eγ−1+βπCt=1. \[\pi _{t}^{K}+\pi _{t}^{H}+\pi _{t}^{C}={{e}^{\gamma -1+{{\beta }_{\pi }}{{K}_{t}}}}+{{e}^{^{\gamma -1+{{\beta }_{\pi }}{{H}_{t}}}}}+{{e}^{\gamma -1+{{\beta }_{\pi }}{{C}_{t}}}}=1.\]

The normalising condition, i.e. Eq. (17), means that 18 eγ−1=1eβπKt+eβπHt+eβπCt. \[{{e}^{\gamma -1}}=\frac{1}{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{H}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{C}_{t}}}}}.\]

Because of Eq. (18), 19 πtK=eβπKteβπKt+eβπHt+eβπCt. \[\pi _{t}^{K}=\frac{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}}{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{H}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{C}_{t}}}}}.\]

In turn, 20 πtH=eβπKteβπKt+eβπHt+eβπCt. \[\pi _{t}^{H}=\frac{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}}{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{H}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{C}_{t}}}}}.\]

And, 21 πtC=eβπKteβπKt+eβπHt+eβπCt. \[\pi _{t}^{C}=\frac{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}}{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{H}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{C}_{t}}}}}.\]

Supposedly, the Labour depends upon the Consumption in the following way: 22 Lt+1=Lmax+Lmin−Lmax1+Ctμ, \[{{L}_{t+1}}={{L}_{\max }}+\frac{{{L}_{\min }}+{{L}_{\max }}}{1+{{\left( {{C}_{t}} \right)}^{\mu }}},\] where L_max is the maximal Labour contribution into the production, possible when 23 Ct→∞; \[{{C}_{t}}\to \infty ;\] L_min is the minimal Labour contribution into the production function, possible when 24 Ct=0; \[{{C}_{t}} = 0;\] μ is the coefficient.

Using Eqs (1)–(24), the recursive system is ascertained as the following: 25 Yt+1Kt+1Ht+1Lt+1Ct+1πt+1Kπt+1H=AKtαHtβLt1−α−βπtKYt+1πtHYt+1Lmax+Lmin−Lmax1+CtμYt+1−Kt+1+Ht+1eβπKteβπKt+eβπHt+eβπCteβπHteβπKt+eβπHt+eβπCt. \[\left( \begin{array}{*{35}{l}} {{Y}_{t+1}} \\ {{K}_{t+1}} \\ {{H}_{t+1}} \\ {{L}_{t+1}} \\ {{C}_{t+1}} \\ \pi _{t+1}^{K} \\ \pi _{t+1}^{H} \\ \end{array} \right)=\left( \begin{array}{*{35}{l}} A{{\left( {{K}_{t}} \right)}^{\alpha }}{{\left( {{H}_{t}} \right)}^{\beta }}{{\left( {{L}_{t}} \right)}^{1-\left( \alpha -\beta \right)}} \\ \pi _{t}^{K}{{Y}_{t+1}} \\ \pi _{t}^{H}{{Y}_{t+1}} \\ {{L}_{\max }}+\frac{{{L}_{\min }}-{{L}_{\max }}}{1+{{\left( {{C}_{t}} \right)}^{\mu }}} \\ {{Y}_{t+1}}-\left( {{K}_{t+1}}+{{H}_{t+1}} \right) \\ \frac{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}}{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{C}_{t}}}}} \\ \frac{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}}{{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{K}_{t}}}}+{{e}^{{{\beta }_{\pi }}{{C}_{t}}}}} \\ \end{array} \right).\]

The accepted data are 26 A=1.872,α=0.3,β=0.3,Lmin=0.5,Lmax=2.5. \[\begin{array}{*{35}{l}} A=1.872, & \alpha =0.3, & \beta =0.3, & {{L}_{\min }}=0.5, & {{L}_{\max }}=2.5. \\ \end{array}\]

The initial conditions are 27 Y0K0H0L0C0π0Kπ0H=1062.51.7951.50.30.3; \[\left( \begin{array}{*{35}{l}} {{Y}_{0}} \\ {{K}_{0}} \\ {{H}_{0}} \\ {{L}_{0}} \\ {{C}_{0}} \\ ^{\pi _{0}^{K}} \\ \pi _{0}^{H} \\ \end{array} \right)=\left( \begin{array}{*{35}{l}} 10 \\ 6 \\ 2.5 \\ 1.795 \\ 1.5 \\ ^{0.3} \\ 0.3 \\ \end{array} \right);\] and 28 μ=1.5,βπ=0.3; \[\begin{matrix} \mu =1.5, & {{\beta }_{\pi }}=0.3; \\ \end{matrix}\]

The results of the computer simulation are shown below (Figs. 1–10):

Figure 1.

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Figure 10.